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Mirrors > Home > MPE Home > Th. List > ax-addcl | Structured version Visualization version GIF version |
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9851. Proofs should normally use addcl 9897 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-addcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 9813 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1977 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1977 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 383 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | caddc 9818 | . . . 4 class + | |
8 | 1, 4, 7 | co 6549 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 1977 | . 2 wff (𝐴 + 𝐵) ∈ ℂ |
10 | 6, 9 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
This axiom is referenced by: addcl 9897 |
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